Chapter 3 Time Value of Money After studying

After studying Chapter 3, you should be able to: Understand what is meant by "the time value of money." Understand the relationship between present and future value. Describe how the interest rate can be used to adjust the value of cash flows – both forward and backward – to a single point in time. Calculate both the future and present value of: (a) an amount invested today; (b) a stream of equal cash flows (an annuity); and (c) a stream of mixed cash flows. Distinguish between an “ordinary annuity” and an “annuity due.” Use interest factor tables and understand how they provide a shortcut to calculating present and future values. Use interest factor tables to find an unknown interest rate or growth rate when the number of time periods and future and present values are known. Build an “amortization schedule” for an installment-style loan.

The Time Value of Money The Interest Rate Simple Interest Compound Interest Amortizing a Loan Compounding More Than Once per Year

Obviously, $10,000 today. You already recognize that there is TIME VALUE TO MONEY!! The Interest Rate Which would you prefer -- $10,000 today or $10,000 in 5 years?

TIME allows you the opportunity to postpone consumption and earn INTEREST. Why TIME? Why is TIME such an important element in your decision?

Types of Interest Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent).

SI = P0(i)(n) = $1,000(.07)(2) = $140 Simple Interest Example Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

FV = P0 + SI = $1,000 + $140 = $1,140 Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (FV) What is the Future Value (FV) of the deposit?

The Present Value is simply the $1,000 you originally deposited. That is the value today! Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (PV) What is the Present Value (PV) of the previous problem?

Why Compound Interest? Future Value (U.S. Dollars)

Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years. Future Value Single Deposit (Graphic) 0 1 2 $1,000 FV2 7%

FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070 Compound Interest You earned $70 interest on your $1,000 deposit over the first year. This is the same amount of interest you would earn under simple interest. Future Value Single Deposit (Formula)

Assume that you need $1,000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 1 2 $1,000 7% PV1 PV0 Present Value Single Deposit (Graphic)

Types of Annuities Ordinary Annuity: Payments or receipts occur at the end of each period. Annuity Due: Payments or receipts occur at the beginning of each period. An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

Hint on Annuity Valuation The future value of an ordinary annuity can be viewed as occurring at the end of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginning of the last cash flow period.

Hint on Annuity Valuation The present value of an ordinary annuity can be viewed as occurring at the beginning of the first cash flow period, whereas the future value of an annuity due can be viewed as occurring at the end of the first cash flow period.

1. Read problem thoroughly 2. Create a time line 3. Put cash flows and arrows on time line 4. Determine if it is a PV or FV problem 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional) Steps to Solve Time Value of Money Problems

1. Solve a “piece-at-a-time” by discounting each piece back to t=0. 2. Solve a “group-at-a-time” by first breaking problem into groups of annuity streams and any single cash flow groups. Then discount each group back to t=0. How to Solve?

General Formula: FVn = PV0(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PV0: PV of the Cash Flow today Frequency of Compounding

Usefulness of Amortization 2. Calculate Debt Outstanding -- The quantity of outstanding debt may be used in financing the day-to-day activities of the firm. 1. Determine Interest Expense -- Interest expenses may reduce taxable income of the firm.